Ernst Abbe came up with a diffraction limit that falls between the Dawes and sparrow limit. In one article the Abbe "limit is referred to as a "hard floor" where by no one can resolve further regardless of scope or observer.

It seems anyway like a more definitive limit than Dawes emperical *notch* values but it doesn't seem to have much practice among amatuers. Anyone know?

Another question I've got is with sparrows limit or non limit as it were. When you really consider it the sparrow limit is arbitrary. Since both light curves of a doublestar diffraction patterns culminate in peaks at the center of the airydisc one pattern would need to be literally overlaying the other in order to fulfill the definition in that that peakswhere merged do not result in a dip in brightness.

It does seem that if there is any condition other than two airy discs right on top of each other, there should be some contrast differential albeit very small and well below the eye's ability to discern it. But, it should be there. Or maybe the light from just off center of each disc adds together and creates a new peak that would have no separation separation from itself - hence no contrast.

Observing 72 Pegasi at below the Sparrow limit, much to my surprise on your suggestion it might be doable, elongation (not resolution) was visible. If two stars show elongation, then their peaks do not coincide and contrast should exist. Of course, 72 Pegasi is not a pair of equal brightness, but it's close. The primary's brighter spurious disc overwhelmed the companion's, so the latter had little chance to provide some contrast to the companion's disc - other than it being dimmer over all.

I think, anyway. But maybe the light from each disc adds to the overall brightness? If so, 72 Peg's companion side of it's airy disc should have been brighter...with the peak shifted from the primary's central airy disc to some point overlapping the companion's disc. But, graphs of Airy discs do not show that behavior.

I dunno, Maybe Sparrow really is not zero contrast at zero separation. But contrast is so very low, while the measure still provides something above zero so line pairs can still be calculated (at something less than diving by zero and an infinite result.)

Thinking about this, say we have two point sources separated by the Raleigh limit. Well, the Airy discs (of 6th mag stars) do overlap. In fact, the edge of one is on the center of the other. So, there is some brightening in the dark space between them, the brightness rises and contrast falls to about 28% in that dark space.

Move them closer to the Dawes limit, and the overlapping dark space contrast decreases further to the eye's limit of 5%. The space between the point sources brightens. Move them closer to the Sparrow limit, same thing happens. The total light in (what would normally be) the dark space increases to the same brightness as the center of the point source itself. Contrast falls to zero at this separation. So, Sparrow is not arbitrary, other than using a standard brightness to calculate it.

Right? So, because the light from each point source does appear to add together, the dark space can brighten to 100% (and contrast to zero) before the point sources become coincident. So, Sparrow will fall to zero contrast a bit before the point sources are on top of one another and there will be some measurable angular separation (hence observable elongation.)

I am not sure what contrast can be expected for Abbe, obviously between 5% (Dawes) and zero (Sparrow) for two 6th mag stars. Since Abbe is non zero, and contrast is resolution, then Abbe can safely be said to be the floor on resolution...if you can actually see the contrast differential like the Dawes limit. You may not see it, but the focal plane can certainly present contrast at the Abbe limit. It stops doing so at Sparrow.

I use Sparrow when thinking MTF because Sparrow IS zero contrast at that spacial frequency (normalized to 1 for a given aperture.)

I have the feeling there are some confusions in the discussion thus far. One might be from attempting to apply ideas and techniques from microscopy to telescope observing - yes, there are some things the same (diffraction theory, for example) but others are different.

I think some of the puzzling comes from trying to deal with visibly non-resolved images of points (stars) that are nevertheless visibly not circular. The Sparrow limit describes the point at which two diffraction discs present an oval or extended appearance without a noticeable dip in brightness between them. There's really no problem over the concept of an oval of uniform brightness that isn't circular (because it's oval).

Asbytec's suggestion that "contrast is resolution" is not true as a generalization. You can "resolve" an oval shape without resolving the components that make it up. And we can get into a detailed (too detailed) discussion of aspects of that.

I'd suggest, for thinking about double stars that can be observed in various forms of sub-resolution, while still obviously not being single, that a useful study is Paul Couteau's chapter on "Optical Concepts Useful to Double-Star Observers" in his book Observing Visual Double Stars. It includes a particularly good section on "Image Structure of a Close Double Star". There's also a table of image appearance/elongation at various fractions of the size of the Airy disk.

A follow-up can be reading Christopher Taylor's section on the appearance of resolved/elongated double stars in his chapter in Bob Argyle's book Observing and Measuring Visual Double Stars (1st or the new 2nd edition).

Fred, sure. You can "resolve" an elongated disc from the surrounding space while not resolving it's components from one another. In other words, I think you're saying, you can tell the star is a double because it's not a circular Airy disc - hence it is resolved as a double.

This is true with 72 Pegasi at 0.5" arc in my 6". And because the companion was slightly dimmer (changing the equation) I could resolve it in the pure sense of seeing a contrast difference. One Airy disc was brighter than the other, that could be seen (actually without much difficulty.) But no contrast could be seen (visually, anyway) in the space where the spurious discs overlap, the primary's disc dominated and appeared equally bright best I could tell. The dimmer companion occupied a tiny bit of space beyond the primary's spurious disc and showed some contrast.

In the pure sense, contrast is resolution and resolution is contrast. If you had two small pieces of paper nearly equally grey, held them close and could not tell them apart, they are not resolved from one another, nor from nearly equally grey surroundings. All Airy discs visible in an aperture are resolved from the dark of space - only at night, of course - even if two are very close together.

I'd like to get both those books actually. Thanks for your points and suggested reading Fred. Norme I still don't see by looking at a diffraction pattern profile of a doublestar how the non contrast could arise unless the peaks were flat topped or plateau like. I don't think you can add light to raise the diffraction profile to fill the contrast dip by combining patterns in a merger. I think the drop is still there in theory of reality but the eyes response has created a contrast plateau between the peaks for lack of the needed sensitivety to see the absolute profile dipthat bridges them together.

If that's the case then a revised profile more in line with the eyes response might be in order.

Pete, at Sparrow that's what happens. The light curve is flat between the peaks. At Dawes, there is a dip between each maximum, but intensity is higher than that same radius from a single point source.

Apparently there is some discrepancy about where Sparrow's limit really is, half the Raleigh limit? Maybe the CO and color play a role at these scales. I'm sticking with the credible sources stating 107/Dmm.

I've got two diagonals. One is a Televue 1.25" diagnal with typical reflective coatings bought the year Hale Bopp blew through. The other is a typical Chinese diagonal that came with my C6 some 8 months ago.

One has easily more light throughput than the other. Which wins? Thick prism or old reflective mirror diagonal?

I think it better to say that resolution requires contrast, not that "resolution is contrast". Obviously when contrast is zero resolution is zero. But if you look at MTF graphs you can see that contrast in differing optical systems will vary at the same resolution.

Likewise, you can print a photo that has a particular resolution level at differing contrast levels. The detail in it doesn't change, though it will look different according to the contrast level.

Dick Suiter has some useful comments on resolution of double stars in Appendix A3 of the 2nd edition of his Star Testing book. Re the Sparrow criterion: "defined as that separation which results in a flat isthmus between the stars. The Sparrow criterion is adjusted for obstructed and aberrated apertures until it always gives that flat region between the stars. Thus, it is always uniquely defined and always delivers the same behavior, but its exact separation varies with details of the aperture". In other words, you can define a particular number for the Sparrow criterion based on perfect optics, but that won't tell you what your imperfect optics will give.

He mentions that for unobstructed apertures (presumably un-aberrated as well) the Sparrow criterion is about 92% of Dawes' criterion or about 77% of Rayleigh's separation. (pages 317-318).

And with regard to exceeding the Sparrow criterion, which Suiter acknowledges some double-star observers have done, "Stars are point objects... Points can be resolved by using the oval shape of the image as the only discriminator". In that sentence "resolved" doesn't mean "separated", rather something closer to "discerned".

An interesting thing about the last paragraph is that it works because it is impossible the oval or elongation to happen any other way barring astigmatism or optical design artifact. The moon by contrast could never allow such "resolution" based on ovular craters. In this respect the interpretation of an elongated star is in effect resolution if you'll accept the assumptions.

Round. Elongated. Notched (aka 'peanut' or 'figure 8'). Black sky split. These should be all anyone needs. These 'limits' - Dawes, Sparrow, Rayleigh, Lord - are, for lack of a better term, limiting. And all of them are so full of asterisks - a whole bunch of "it depends". Will someone chose not to observe a pair because it lies below an arbitrary limit? I hope not. Pour on the X's and enjoy.

Round. Elongated. Notched (aka 'peanut' or 'figure 8'). Black sky split. These should be all anyone needs. These 'limits' - Dawes, Sparrow, Rayleigh, Lord - are, for lack of a better term, limiting. And all of them are so full of asterisks - a whole bunch of "it depends". Will someone chose not to observe a pair because it lies below an arbitrary limit? I hope not. Pour on the X's and enjoy.

Dave

I generally go with "Unresolved", "Elongated" (no notching), "Resolved" (clear notching, but spurious disk overlap with no dark sky between the disks), and "Split" (fully resolved with at least a thin lane of dark sky visible). Dawes limit can be a useful thing to get a very rough idea of what doubles might be worth looking at in a given aperture, but much beyond that, all one needs to do is to actually observe the double and see if its duplicity can be detected. Clear skies to you.

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Will someone chose not to observe a pair because it lies below an arbitrary limit?

Point well taken, until Pete urged me to break the Sparrow limit, I never would have succeeded in "discerning" 72 Pegasi as a double. Probably would have written it off as more appropriate for a 10" scope. It's combined spurious discs were clearly elongated at 0.5" arc, and even more easily noticeable in the elongated diffraction ring.

Pete, lets say that Sparrow indeed has a flat contrast between two points of 6th mag. What would happen if you lowered the magnitude to the limit of the aperture. Surely the point sources would become dim and small, the suspicious discs would decrease in radius. Would that lack of contrast between them persist? Seems it should, but probably not. So there should be some dark space and resolution as two distinct points. Right?

Okay, if so, then we should be able to move those very faint and tiny point sources closer together until contrast between them again falls off to zero. At the limiting magnitude of an aperture, that would seem to be the real limiting resolution.

Now, to see a dim star, of course it has to contrast against the blackness of space. So, lets say that contrast is consistent with what the average eye can detect at about 5%. I don't know the magnitude that would correspond to off hand, so let's guess it means we can see the top 5% of the Airy intensity. I am trying to tie this in with the Airy disc and the size of the spurious disc (the percentage of light seen at a given magnitude.)

A bright star puts about 85% of it's light into the spurious disc, a 6th magnitude star about 50 to 60% - meaning the spurious disc is about half the Airy disc radius. Following that logic, 5% of the light would be seen in the spurious disc of our dimmest star. This would give a very dim star's radius for a 150mm aperture of (0.92" * 0.05 = 0.046" arc.)

Nudging these two dim point sources together so their discs just touched leaving zero contrast (any dimmer and neither the disc nor the dark space could be seen) would yield a separation of 0.092" arc. Interestingly, that is about 1/10th the Raleigh limit. (Not sure if that would apply across all apertures, maybe.)

Okay, now for the fun part. A separation of 0.092" arc yields a frequency of ~ 10.9 line pairs per arc second. At this frequency, you could squeeze nearly 11 of these double stars into an arc second equating to (60 * 10.9 = 652) line pairs/arc min on the focal plane with 5% contrast (and 100% target contrast.)

There is no way the eye can resolve this frequency, so you have to magnify it to reduce the frequency. If you magnified it 652x (~110x per inch) you would have one line pair per apparent arc minute - about the resolution of the eye. At that angular size, you should be able to see one line pair with normal 20/20 vision.

So, how crazy is that? I kind of like it, actually. Needs refinement.

There are other variables when discussing human intervention, and also properties of the optical train including CO and light loss, etc. We could incorporate Suiter's EER concept which includes the obstruction effects. Maybe later. Except for the 5% contrast level and assuming 20/20 vision, the rest have been conveniently ignored and just working from the spurious disc the human eye might be able to see. Terms like dark space and resolution kind of imply the existence of an observer to even notice it.

If no one has thought of it before, I would hereby dub this limit (including further revisions to theory) the Azure Limit, in honor of the guy's online avatar who inspired it.

I think in seeking out the dimming between what might appear to be a Sparrow double would be an interesting exercise. I think the potential fly in the ointment ( or pattern as it were) is that while these dimmer diffraction patterns have a smaller apparent spurious disc the eyes sensitivety at these lower levels becomes grainy to even peripheral vision needy. A 3" double at 11.5v for example is actually kinda difficult. Totally doable but it'd morph kinda blob like from singular to double in this way that was shadowy and ambiguous. By contrast if it were 7v it'd be a walk in the park in the same seeing. I couldn't benefit from higher apparent resolution between spurious discs because ybrud appear to begin to dissolved. For me I think 9v is my comfortable putter edge for this kinda thing.

I appreciate the points brought here by ev3ryone. Hey reversing my cooling fan and tweaking the system again. Gonna move the boundary Fab lower over the primary. Tonight's a mix and I'm taking the night off so ill hit some doubles. Hope the seeing is cooperative.

I use Dawes as lower limit when selecting doubles from the WDS catalogue for a session - it is hard enough when seeing is not very good and I prefer anyway to have my doubles clear split with some dark space in between or at least an 8 so I prefer rather Rayleigh limit separations. Elongations are not visually attractive to me and I count them as split only if there is a clear (and afterwards confirmed) indication of the position of the companion.Does anybody know of the error range of Dawes limit because there must one be as it is based on statistical analysis of empirical data? To give limits as single numbers without an error range gives the impression of accuracy which does not exist. Even for the Rayleigh you have to consider a +/-10% range depending on the spectrum wavelength.Wilfried

Wil you raise a good point. There does have to be some range of error.

Elongations CA can be very interesting when tyhe overlap of patterns is different colors. It adds a very compelling aspect. Ideally I'd love to catch a carbon star elongated with a blue star. That would be really engaging. To date they have been equal or near so with the b component of gamma andromeda. when it was elongated showed a wild color contrast of vivid blue to a dimmer smaller ashen white. That's been the one to beat. Seeing a carbon component at 0.4 or less would be amazing.

Maybe I can retry gamma tonight but its so so tight now and troublingly uneven disc sizes makes it a slimmer chance that its be component would be leaking out the side anymore.

Oh the nomenclature - lol- I vote Aldrin Limit based on my avatar. A man whose attained the highest and fallen the hardest only to come up again.

You know what the Azure Limit is? The point where despite being exhausted I still defiantly set up and observe. That's my method all too often.

You know, when we resolve stars, we're really not resolving one from the other. Especially with evenly bright pairs that do not vary in contrast from one another, there is no resolution of one from the other by definition. We're really resolving each component star from the blackness of space where the contrast does exists.

So, maybe you do not need a dark space between them to call resolution. An elongation has just enough black space around it to show a close companion. Maybe the companion does not need black space all the way around it, just enough to set it apart from the primary. But, that's because we know diffraction patterns are circular.

Imagine two white, square cards on a black background. Push them together and you cannot tell whether there is one rectangle, two squares, or even 4 triangles. It looks like one rectangle, but could be any number of unresolved shapes. But, if we know they are squares, then there is only one resolution as to what we're seeing: two squares side by side.

In fact, we could push them to overlap to such an extent as long as it was not square we could detect the presence of at least two squares or even 35 unresolved squares tightly packed. But, such an appearance would show more than one white square must exist in the image. But we know this because the white barely rectangular shaped are set against a contrasting backdrop, not from one another.

I see where you are going and it would seem interpttation is a kind of resolution in and of itself. Its not but shifting the emphasis could make it seem such. Rayleigh is concrete and Dawes is an arbitrary agreement of a resolved state of merged patterns for the sake of a common threshold or benchmark by which to classify a specific split per an apertures Res. he could have equally said " well its the Dawes limit the moment elongation is seen at all". in pactice this was probably far to taxing to perform reliably given real world seeing and optical imperfections. Backing off a bit and having the first pinch reveal I guess seemed less an Olympic achievement and more reliable a feature to repeatedly gauge with certain measure. Thomas Jensen elongated. 50 Dawes with an 80mm which is a commendable feat but probably not repeatabley certain as the notch. All that to say I think where they drew the line empericaly was alltigood end even if a "limit" could've been pushed further. Of ourselves Norme the ratio to apertures never change even tho it could look like an optic was stretching.

I'm still not settled on either Abbe or Sparrow as they seem go break down the finer you cut the wire. I RESPECT them but I've got issues. Then again what lines of law emperical measure or established Rayleigh doesn't challenge its own definition upon close enough examination ? move a hundreth past the Rayleigh for an 80mm - who would ever know but by the measure it was breeched.

Sure, Raleigh is concrete in that it defines the Airy disc under specific conditions such as the wavelength and aperture. It's still possible not to technically resolve, being a clean dark space, between two brighter stars. At some point, the spurious discs will over lap with no detectable contrast.

Dawes was more empirical based on observation. But, they still define resolution and placing a definition or conditions on resolution is key. If we define it differently, we can "resolve" stars far closer together.

By the way, I discovered it's impossible to resole unequal pairs using the Aldrin limit. When they are that close together, the brighter spurious disc completely overwhelms the companion. It can only be split at the limiting magnitude or by elongation of an equally bright companion. But, the line pairs should be visible under the best conditions and if the spurious discs are literally pin points. And in some sense, that should be the limit except for some difficult to measure change in the Airy pattern as two stars are literally on top of one another.

But, oh well, 72 Peg was not as difficult as I had imagined based on Sparrow's limit. Cool! Still not sure what you have against Abbe limit. Maybe it's some practical limit.

You know what it is, Dawes has a specific look that's easier than Abbe. I think its easier to call a Dawes than an Abbe in the field. Yeah so in a real way with reservations a limit is what you make it. I guess that was my point condensed into a sentence.